Expansion systems

TL;DR

This thesis introduces expansion systems as a versatile framework for various approximation algorithms, analyzing their properties, convergence criteria, and isomorphisms, with some convergence claims still unproven.

Contribution

It provides a unified theoretical framework for approximation methods like Taylor, decimal, and continued fractions, including new concepts such as isomorphisms.

Findings

Basic properties of expansion systems are established.

Criteria for convergence are studied.

Introduction of isomorphisms between expansion systems.

Abstract

In this master's thesis, we introduce expansion systems as a general framework to describe a large variety of approximation algorithms, such as Taylor approximation, decimal expansion and continued fraction. We consider some basic properties of expansion systems, and also study criteria for convergence. Further, we introduce the notion of isomorphisms between expansion systems. In the appendix, we discuss another class of expansion systems, which we call approximation systems. Many claims of convergence in this appendix, remain to be proven.